Strong Stability for the Wulff Inequality with a Crystalline Norm

Let K be a convex polyhedron and F its Wulff energy, and let C (K) denote the set of convex polyhedra close to K whose faces are parallel to those of K. We show that, for sufficiently small epsilon, all epsilon-minimizers belong to C (K). As a consequence of this result we obtain the following sharp stability inequality for crystalline norms: There exist gamma = gamma(K,n) > 0 and sigma = sigma (K,n) > 0 such that, whenever vertical bar E vertical bar = vertical bar E vertical bar and vertical bar E Delta K vertical bar <= sigma, then F(E) - F(K-a) >= gamma vertical bar E Delta K-a vertical bar for some K-a is an element of C (K). In other words, the Wulff energy F grows very fast (with power 1) away from C (K). The set K-a is an element of C (K) appearing in the formula above can be informally thought as a sort of projection of E onto C (K). Another corollary of our result is a very strong rigidity result for crystals: For crystalline surface tensions, minimizers of F (E) integral(E) g with small mass are polyhedra with sides parallel to the those of K. In other words, for small mass, the potential energy cannot destroy the crystalline structure of minimizers This extends to arbitrary dimensions a two-dimensional result obtained in [9]. (C) 2020 Wiley Periodicals LLC

Automated summary

This paper investigates the relationship between the Wulff energy of a convex polyhedron and its approximations, proves partial results, and provides related stability inequalities and rigidity results.